The Physics Philes, lesson 39: Angling to Get Ahead

In which bodies are rigid, circles are drawn, and degrees are converted.

Okey dokey, people. We just finished our discussion of momentum. It was glorious, but it’s time to move on. We have bigger fish to fry. And by “fish” I mean “topics” and by “to fry” I mean “to learn about.” Specifically, today we start our discussion on the rotation of rigid bodies.

You may recognize the term “rigid bodies.” I used it last week without defining it. Did you catch it? Where you pulling your hair out, muttering, “What is a rigid body? What does it mean? WHAT DOES IT MEAN?!?” Well never fear! Today is the day I tell you what it means to have a rigid body.

I may have built it up too much. A rigid body is just an idealized model of a body that is perfectly definite and has an unchanging size and shape. It’s like when we calculate the velocity and acceleration of a body going in a straight line. We assume it’s a mere point on a plane. With angular velocity and acceleration, we pretend that the bod is rigid so we don’t have to deal with the squishy and floppiness of real life.

Angular motion is rotational motion, so when analyzing this type of problem we envision the rigid body rotating around a fixed axis. All a fixed axis is an axis that doesn’t move in an inertial frame of reference and doesn’t change direction relative to that frame. On an xy-plane, we add a z-coordinate that pokes out from the origin. If you fell on it, it would stab you through the heart and you would die. Think about the speedometer on your car. That line or arrow that indicates your speed is rotating around a fixed axis. It has angular motion.

To analyze problems using angular velocity or angular acceleration, we need to know angles. We use the Greet letter Θ to represent the angular coordinate. This coordinate could be positive or negative, depending on what direction it’s going around the coordinate plane.

So we describe rotational motion using angles instead of (x,y) coordinates. But there is another wrinkle. We don’t use degrees; we use radians. A radian is the angle subtended at the center of a circle by an arc the length equal to the radius of the circle.

Whaaaaaat? Here’s a picture:

The blue is the arc and r is the radius of the circle. So one radian is the angle you get when you draw a line from the center of the circle that creates an arc length that is equal to radius. I’m not sure those words make it any clearer…

Anyhoo, we can use the relationship between the radius, the arc length, and Θ to our advantage using math! To find the arc length, all we need to do is multiply Θ by the radius: s = rΘ.

Be careful to always use radians with this equation. If you use degrees, you will get very, very incorrect answers. Trust me on this. I know from experience.

Because you need to know an angle in radians, you need to know how to convert degrees to radians. To convert degrees to radians, all you have to do is multiply the number of degrees in question by Π/180°. For example, say we have an angle of 45°. We would multiply 45°(Π/180°) = Π/4 rad. Some other common conversions are:

30° = Π/6

60° = Π/3

90° = Π/2

This makes sense when you think of a circle. A circle has a circumference of 2Π times the radius, so 2Π is the number of radians in a circle. Which means that a semi-circle has Π radians, or 180°. When you split 180 in half, you get 90, or Π/2; when you split it into thirds you get 60; when you split it into sixths you get 30.

Clear as mud, right? This values may or may not be useful in the practice problems we’ll try, but they are handy to have off the top of your head.

Now you have a foundation for discussing angular velocity and acceleration. But we won’t get to that today. Next week we’ll learn how to calculate angular velocity and acceleration of rigid bodies. See you then!